On the Real-rootedness of the Descent Polynomials of $(n-2)$-Stack Sortable Permutations

B\'ona conjectured that the descent polynomials on $(n-2)$-stack sortable permutations have only real zeros. Br\"and\'en proved this conjecture by establishing a more general result. In this paper, we give another proof of Br\"and\'en's result by using the theory of $s$-Eulerian polynomials recently developed by Savage and Visontai.


Introduction
Suppose that w = w 1 · · · w n is a permutation of a set of distinct numbers and w i is the maximal number of {w 1 , . . . , w n }. The stack sorting operation s on w can be recursively defined as s(w) = s(w 1 · · · w i−1 )s(w i+1 · · · w n )w i .
For σ = (σ 1 , σ 2 , . . . , σ n ) ∈ S n , let denote the set of descents of σ, and let des σ = | Des σ|. The Eulerian polynomials A n (x) are usually defined as the descent generating function over S n , namely, Let W t (n, k) be the number of t-stack sortable permutations in S n with k descents, and let W n,t (x) = n−1 k=0 W t (n, k)x k be the descent polynomials over t-stack sortable permutations. Bóna [1] showed that for fixed n and t the descent polynomial W n,t (x) is symmetric and unimodal, and proposed the following conjecture.
). The descent polynomial W n,t (x) has only real zeros for any integer 1 ≤ t ≤ n − 1.
The above conjecture is true for t = 1, 2, n − 2, or n − 1, see Brändén [2] and references therein. In fact, W n,1 (x) are the Narayana polynomials and W n,n−1 (x) are the Eulerian polynomials, both of which are known to be real-rooted. Based on a compact and simple form of W 2 (n, k) due to Jacquard and Schaeffer [6], Brändén proved the real-rootedness of W n,2 (x) by using the tool of multiplier sequences. For t = n − 2, it is easy to show that By using certain real-rootedness preserving linear operators, Brändén proved the realrootedness of W n,n−2 (x). Remarkably, Brändén [2] obtained the following result.
). For any n ≥ 3 and k ≥ −2, the polynomial has only real zeros.
The main objective of this paper is to give another proof of the above result by using the theory of s-Eulerian polynomials recently developed by Visontai and Savage [10]. The s-Eulerian polynomials have proven to be a powerful tool for studying the real-rootedness of Euerlian-like polynomials, see also Yang and Zhang [14]. Instead of directly proving Theorem 1.2, we shall prove a slightly general result as shown below. Theorem 1.3. For any n > 3 and k ≥ −n, the polynomial A n (x) + kx A n−2 (x) has only real zeros.
The remainder of the paper is organized as follows. In Section 2, we shall give a brief overview of the theory of s-Eulerian polynomials and related results. In Section 3, we shall give a proof of Theorem 1.3. In Section 4, we shall present one open problem.

The s-Eulerian polynomials
The aim of this section is to review some terminology and results on s-Eulerian polynomials.
Let s = (s 1 , s 2 , . . .) be a sequence of positive integers. Following Savage and Visontai [10], we say that an n-dimensional s-inversion sequence is a sequence e = (e 1 , . . . , e n ) ∈ N n such that e i < s i for each 1 ≤ i ≤ n. Let I Let asc e = | Asc e|, and let Savage and Visontai [10] called E It is clear that The key point is that these polynomials satisfy certain simple recurrence relation and certain mutually interlacing property. Let us first recall the definition of mutually interlacing. Given two real-rooted polynomials f (z) and g(z) with positive leading coefficients, let {r i } be the set of zeros of f (z) and {s j } the set of zeros of g(z). We say that g(z) interlaces f (z), denoted g(z) f (z), if either deg f (z) = deg g(z) = n and s n ≤ r n ≤ s n−1 ≤ · · · ≤ s 2 ≤ r 2 ≤ s 1 ≤ r 1 , or deg f (z) = deg g(z) + 1 = n and r n ≤ s n−1 ≤ · · · ≤ s 2 ≤ r 2 ≤ s 1 ≤ r 1 .
We say that a sequence of real polynomials (f 1 (x), . . . , f m (x)) with positive leading coefficients is mutually interlacing if f i (x) f j (x) for all 1 ≤ i < j ≤ m. Savage and Visontai obtained the following result.
We shall also need the following result due to Haglund, Ono, and Wagner [5].
It is known that interlacing of two polynomials implies the real-rootedness of their arbitrary linear combination, see Obreschkoff [8] and Dedieu [4]. have only real zeros for any real numbers α and β with α 2 + β 2 = 0. Therefore, we have the following result.
has only real zeros.

Proof of Theorem 1.3
In this section we aim to give a proof of Theorem 1.3. Note that A n−2 (x) = A n−1,0 (x).
Before proving Theorem 1.3, let us first express A n (x) in terms of A n−1,i (x). Lemma 3.1. For any integer n ≥ 2, we have Proof. By (12), we obtain that Then, by interchanging the order of summation for each double summation, we get that which leads to the desired equality. This completes the proof.
Now we proceed to give a proof of our main theorem.
Proof of Theorem 1.3. By (13), we have In view of that A n−1,n−2 (x) = xA n−1,0 (x), we have Now we shall use Corollary 2.6 to obtain the real-rootedness of K n (x). To this end, let m = n − 1, f i (x) = A n−1,i−1 (x) for 1 ≤ i ≤ m and It is routine to check that f i (x), a i and b i satisfy the conditions of Corollary 2.6. This completes the proof.

One open problem
We have shown that for any n > 3 and k ≥ −n, the polynomial K n (x) in (2) has only real zeros. Stanley [11] advised us to further study under what conditions does the polynomial K n (x) have only real zeros.
Let T n be the n-th tangential or "Zag" number, see [12, A000182] and let a(n) = T n+1 /T n . Computer evidence suggests the following conjecture.
Note that there is a useful criterion for determining whether a polynomial of degree n has n distinct real zeros. Suppose that These determinants are known as the Hurwitz determinants of f (x) and g(x). Hermite showed that the real-rootedness of f (x) can be uniquely determined by the signs of ∆ 2k (f (x), f ′ (x)). The following is essentially due to Borchardt and Hermite [9, pp. 349].
Using this characterization, we have verified that, for 3 ≤ n ≤ 18, the polynomial K n (x) has only real zeros when k ≤ −n(n − 1) and k ≥ −a(⌊n/2⌋).